[PPT] - Implementing the three-particle quantization condition: a progress PowerPoint Presentation

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S. Sharpe, ``Progress on implementing the three-particle quantization condition,” FLQCD19,

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Implementing the three-particle quantization condition: a progress report

Steve Sharpe University of Washington

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Max Hansen & SRS: “Relativistic, model-independent, three-particle quantization condition,” arXiv:1408.5933 (PRD) “Expressing the 3-particle finite-volume spectrum in terms of the 3-to-3 scattering amplitude,” arXiv:1504.04028 (PRD) “Perturbative results for 2- & 3-particle threshold energies in finite volume,” arXiv:1509.07929 (PRD) “Threshold expansion of the 3-particle quantization condition,” arXiv:1602.00324 (PRD) “Applying the relativistic quantization condition to a 3-particle bound state in a periodic box,” arXiv: 1609.04317 (PRD) “Lattice QCD and three-particle decays of Resonances,” arXiv: 1901.00483 (to appear in Ann. Rev. Nucl. Part. Science)

2

3-particle papers

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Raúl Briceño, Max Hansen & SRS: “Relating the finite-volume spectrum and the 2-and-3-particle S-matrix for relativistic systems of identical scalar particles,” arXiv:1701.07465 (PRD) “Numerical study of the relativistic three-body quantization condition in the isotropic approximation,” arXiv:1803.04169 (PRD) “Three-particle systems with resonant sub-processes in a finite volume,” arXiv:1810.01429 (PRD) Tyler Blanton, Fernando Romero-López & SRS: “Implementing the three-particle quantization condition including higher partial waves,” arXiv:1901.07095 (JHEP) SRS “Testing the threshold expansion for three-particle energies at fourth order in φ4 theory,” arXiv:1707.04279 (PRD) Tyler Blanton, Raúl Briceño, Max Hansen, Fernando Romero-López, SRS &Adam Szczepaniak: works in progress

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Outline

Motivations for studying 3 (or more) particles
Status of theoretical formalism for 2 and 3 particles
Numerical implementation of 3-particle QC
Isotropic approximation
Including higher partial waves
Isotropic approx. v2: including two-particle bound states
Outlook

4

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Motivations for studying three (or more) particles using LQCD

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Studying resonances

6

N.B. If a resonance has both 2- and 3-particle strong

decays, then 2-particle methods fail—channels cannot be separated as they can in experiment

Most resonances have 3 (or more) particle decay channels
(no subchannel resonances)
Roper: (branching ratio 25-50%)
(studied by HALQCD—talk by Ikeda)

ω(782, IGJPC = 0−1−−) → 3π a2(1320, IGJPC = 1−2++) → ρπ → 3π

N(1440) → Δπ → Nππ

X(3872) → J/Ψππ Zc(3900) → πJ/ψ, ππηc, ¯ DD*

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Weak decays

Calculating weak decay amplitudes/form factors

involving 3 particles, e.g. K→πππ

7

N.B. Can study weak K→2π decays independently of

K→3π, since strong interactions do not mix these final states (in isospin-symmetric limit)

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A more distant motivation

8

CERN-EP-2019-042 LHCb-PAPER-2019-006 March 21, 2019

CERN-EP-2019-042 13 March 2019

Observation of CP violation in charm decays

LHCb collaboration†

Abstract A search for charge-parity (CP) violation in D0 ! K−K+ and D0 ! π−π+ decays is reported, using pp collision data corresponding to an integrated luminosity

f 6 fb−1 collected at a center-of-mass energy of 13 TeV with the LHCb detec-
tor. The flavor of the charm meson is inferred from the charge of the pion in

D∗(2010)+ ! D0π+ decays or from the charge of the muon in B ! D0µ−¯ νµX decays. The difference between the CP asymmetries in D0 ! K−K+ and D0 ! π−π+ decays is measured to be ∆ACP = [18.2 ± 3.2 (stat.) ± 0.9 (syst.)] ⇥ 10−4 for π-tagged and ∆ACP = [9 ± 8 (stat.) ± 5 (syst.)] ⇥ 10−4 for µ-tagged D0 mesons. Combining these with previous LHCb results leads to ∆ACP = (15.4 ± 2.9) ⇥ 10−4, where the uncertainty includes both statistical and systematic contributions. The measured value differs from zero by more than five standard deviations. This is the first observation of CP violation in the decay of charm hadrons.

5.3σ effect

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A more distant motivation

9

Calculating CP-violation in D→ππ, KK̅ in the Standard Model
Finite-volume state is a mix of 2π, KK̅, ηη, 4π, 6π, …
Need 4 (or more) particles in the box!

weak strong

D 2π 4π

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3-body interactions

10

Determining NNN interaction
Input for effective field theory treatments of larger nuclei & nuclear matter
Similarly, πππ, πKK̅, … interactions needed for study
f pion/kaon condensation

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Dudek, Edwards, Guo & C.Thomas [HadSpec], arXiv:1309.2608

3mπ

LQCD spectrum already includes 3+ particle states

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YITP , 4/25/2019 5 10 15 20 25 30 35 40 45 Level 2.5 3.0 3.5 4.0 4.5 5.0

E mK

single-hadron dominated two-hadron dominated significant mixing π (0) π (2) η (2) π (0) π (0) π (2) π (2)

energies I = 1, S = 0, T+

2u channel

LQCD spectrum already includes 3+ particle states

12

Slide from seminar by Colin Morningstar, Munich, 10/18

4mπ

r mπ ∼ 240 MeV

two-meson operators

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Status of theoretical formalism for 2 & 3 particles

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The fundamental issue

Lattice simulations are done in finite volumes;

experiments are not

14

How do we connect these?

?

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Lattice QCD can calculate energy levels of multiparticle

systems in a box

How are these related to infinite-volume scattering

amplitudes (which determine resonance properties)?

15

iMn→m

Discrete energy spectrum Scattering amplitudes

E0(L) E1(L) E2(L)

The fundamental issue ?

N . B . T h i s i s a fi n i t e v

l

u m e Q F T p r

b

l e m ( c a n i g n

r

e l a t t i c e s p a c i n g )

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When is the spectrum related to scattering amplitudes?

L<2R No “outside” region. Spectrum NOT related to scatt. amps. Depends on finite-density properties

✘

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When is the spectrum related to scattering amplitudes?

[Lüscher]

✔

L

L>2R There is an “outside” region. Spectrum IS related to scatt. amps. up to corrections proportional to e−MπL Theoretically understood; numerical implementations mature. L<2R No “outside” region. Spectrum NOT related to scatt. amps. Depends on finite-density properties

✘

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…and for 3 particles?

18

Spectrum IS related to 2→2, 2→3 & 3→3

scattering amplitudes up to corrections proportional to e−ML [Polejaeva & Rusetsky]

Formalism developed in a generic

relativistic EFT [Hansen & SRS, Briceño, Hansen & SRS]

Alternative approaches based on NREFT

[Hammer, Pang & Rusetsky] and on ``finite- volume unitarity” [Döring & Mai] under development (reviewed in [Hansen & SRS])

HALQCD approach can be extended to 3

particles in NR domain [Aoki et al.]

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Reminder of 2-particle quantization condition

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Two particles (say pions) in cubic box of size L with PBC and total momentum P
Below inelastic threshold (4 pions), the finite-volume spectrum E1, E2, ... is given

by solutions to a equation in partial-wave (l,m) space (up to exponentially suppressed corrections)

K2~tan δ/q is the K-matrix, which is diagonal in l,m
FPV is a known kinematical “zeta-function”, depending on the box shape & E;

It is off-diagonal in l,m, since the box violates rotation symmetry [Lüscher 86 & 91; Rummukainen & Gottlieb 85; Kim, Sachrajda & SRS 05; …]

Single-channel 2-particle quantization condition

det ⇥ (F f

PV)−1 + K2

⇤ = 0

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Single-channel 2-particle quantization condition

Infinite-dimensional determinant must be truncated to be practical;

truncate by assuming that K2 vanishes above lmax

If lmax=0, obtain one-to-one relation between energy levels and K2

det ⇥ (F f

PV)−1 + K2

⇤ = 0

“measured” energy-level CM energy

E∗

n =

q E2

n − ~

P 2

K2,s(E∗

n) = −

1 F f

PV;00;00(En, ~

P, L)

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[Dudek, Edwards & Thomas, 1212.0830]

L/a

Can reconstruct phase shift, which exhibits ρ resonance

a E*

Application to ρ meson

mπ ≈ 400 MeV

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S-wave above 2mπ, 2mK, and 2m𝜃 Ansatz

57 energy levels

χ2/Ndof = 44.0 57 − 8 = 0.90

0.2 0.4 0.6 0.8 0.14 0.16 0.18 0.20 0.22 0.24 0.2 0.14 0.16 0.18 0.20 0.22 0.24

~ “cross section”

K−1(s) =   a + bs c + ds e c + ds f g e g h  

[Briceño, Dudek, Edwards, & Wilson arXiv:1708.06667] mπ=391 MeV

State-of-the-art: coupled channels

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[Briceño, Dudek, Edwards & Wilson arXiv:1708.06667]

State-of-the-art: coupled channels

0.06
0.04
0.02

0.02 0.04 0.06 0.12 0.14 0.16 0.18 0.20 0.22 0.24

Parametrization dependence of pole positions

mπ = 391 MeV

f0

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3-particle quantization condition

25

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Two-step method

26 26

Quantization conditions 2 & 3 particle spectrum from LQCD Integral equations in infinite volume Intermediate, unphysical scattering quantity

det [F−1

2

+ 𝒧2]

det [F−1

3

+ 𝒧df,3]

Scattering amplitudes

ℳ22 , ℳ23 , ℳ32 , ℳ23

L

= 0 = 0

N e e d f

r

t w

s

t e p s c

m

m

n

t

a

l l a p p r

a

c h e s ( t h

u

g h i n t e r m e d i a t e q u a n t i t i e s d i f f e r )

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Meaning of quantization condition

27 27

det [F−1

3

+ 𝒧df,3]

F3 ≡ F 2ωL3

− 2

3 þ 1 1 þ ½1 þ K2G−1K2F

−

F = G =

All quantities are infinite-dimensional matrices with indices describing 3 on-shell particles

ˆ a∗ − → `, m

(E − !k, ~ P − ~ k) (!k,~ k)

BOOST

[finite volume “spectator” momentum: k=2πn/L] x [2-particle CM angular momentum: l,m]

F (closely related to FPV) and G are known kinematic functions depending on L and E
F3 contains effects of two-particle scattering, entering through K2
For large spectator-momentum k, the other two particles are below threshold; we

must include such configurations by analytic continuation up to a cut-off at k~m

= 0

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Status of relativistic approach

28 28

det [F−1

3

+ 𝒧df,3]

Original work applied to scalars with G-parity & no subchannel

resonances [Hansen & SRS, arXiv:1408.5933 & 1504.04248]

E0(L) E1(L) E2(L)

Kdf,3

M3

= 0 UPDATE: subchannel resonances may be allowed by adopting a variant of the PV pole-prescription

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Status of relativistic approach

29 29

Second major step: removing G-parity constraint, allowing 2↔3

processes [Briceño, Hansen & SRS, arXiv:1701.07465]

det ( F2 F3)

−1

+ ( 𝒧22 𝒧23 𝒧32 𝒧df,33) = 0

F2 appears in 2-particle quantization condition

E0(L) E1(L) E2(L)

M22 M23 M32 M33

Kdf,e

2e 2

Kdf,3e

2

Kdf,e

23

Kdf,33

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Final major step: allowing subchannel resonance (i.e. pole in K2)

[Briceño, Hansen & SRS, arXiv:1810.01429]

det ( F˜

2˜ 2

F˜

23

F3˜

2

F33)

−1

+ ( 𝒧df,˜

2˜ 2

𝒧df,˜

23

𝒧df,3˜

2

𝒧df,33) = 0

resonance + particle channel (not physical, but forced on us by derivation) Determined by K2 & Lüscher finite-volume zeta functions

E0(L) E1(L) E2(L)

Kdf,e

2e 2

Kdf,3e

2

Kdf,e

23

Kdf,33

M3

Status of relativistic approach

No unphysical channel in final scattering amplitude

UPDATE: this elaboration may be avoidable

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Numerical implementation: isotropic approximation

31

[Briceño, Hansen & SRS, arXiv:1803.04169]

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Overview

32 32

det [F−1

3

+ 𝒧df,3]

E0(L) E1(L) E2(L)

Kdf,3

M3

= 0 DREAM:

LQCD determine predict

Integral equations

E0(L) E1(L) E2(L)

Kdf,3

M3

REALITY:

fit parametrize predict

TODAY:

predict

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Truncation

33 33

To use quantization condition, one must truncate matrix space, as

for the two-particle case

Spectator-momentum space is truncated by cut-off function H(k)
Need to truncate sums over l,m in K2 & Kdf,3

det [F−1

3

+ 𝒧df,3]

[finite volume “spectator” momentum: k=2πn/L] x [2-particle CM angular momentum: l,m]

matrices with indices:

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Isotropic low-energy approximation

34 34

Scalar particles with G parity so no 2⟷3 transitions and no

subchannel resonances (e.g. 3 π+)

2-particle interactions are purely s-wave, and determined by the

scattering length, a, alone

Avoiding poles in K2 restricts scattering length to 1 > a > −∞, implying no

two-particle bound states

Point-like three-particle interaction Kdf,3 independent of momenta,

although can depend on s=(Ecm)2

Consider only P=0 (though formalism applies for all P)
Analog in our formalism of the approximations used in other approaches:

[Hammer, Pang, Rusetsky, 1706.07700; Mai & Döring, 1709.08222; Döring et al., 1802.03362; Mai & Döring, 1807.04746] [Hansen & SRS]

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Isotropic low-energy approximation

35 35

[Briceño, Hansen & SRS, 1803.04169]

1/Kiso

df,3(E∗) = −F iso 3 [E, ~

P, L, Ms

2]

M3(E⇤, Ω0

3, Ω3) = S

" D + L 1 1/Kiso

df,3 + F iso 3,1

R #

Relation of Kdf,3 to M3 (matrix equation that becomes integral equation when L→∞)

det [F−1

3

+ 𝒧df,3]

L→∞ limit of F3iso depends on M2 & kinematical factors symmetrization D, L & R depend

n M2 &

kinematical factors

M3

½Fs

3kp ¼ 1

L3 ˜ Fs 3 − ˜ Fs 1 1=ð2ωKs

2Þ þ ˜

Fs þ ˜ Gs ˜ Fs

kp

¼ Fiso

3 ðE; LÞ ¼ h1jFs 3j1i ¼

X

k;p

½Fs

3kp

= 0

Reduces problem to 1-d quantization condition, with intermediate matrices involve

finite-volume momenta up to cutoff |k|~m

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Solutions with Kdf,3=0

36 36

iM2 iM2 iM2 iM2 iM2

+ + · · · iM3 = S 

Useful benchmark: deviations measure impact of 3-particle interaction
Caveat: scheme-dependent since Kdf,3 depends on cut-off function H
Qualitative meaning of this limit for M3:

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Solutions with Kdf,3=0

37 37

4 5 6 7 8

mL

3.0 3.5 4.0 4.5 5.0

En(L)/m a = −1/2

(2,2,0) (2,1,1)

These two states are degenerate in the NR theory

Non-interacting states

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Solutions with Kdf,3=0

38 38

4 5 6 7 8

mL

3.0 3.5 4.0 4.5 5.0

En(L)/m a = −1/2

Weakly attractive two-particle interaction

1/L expansion

m

[Beane, Detmold, Savage; Tan; Hansen & SRS]

2-particle interaction enters at 1/L3, 3-particle interaction (and relativistic effects) enter at 1/L6

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Solutions with Kdf,3=0

39 39

4 5 6 7 8

mL

3.0 3.5 4.0 4.5 5.0

En(L)/m a = −1/2

Strongly attractive two-particle interaction

4 5 6 7 8

mL

3.0 3.5 4.0 4.5 5.0

En(L)/m a = −10 m

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Impact of Kdf,3

40 40

4.0 4.5 5.0

mL

2.50 2.55 2.60 2.65 2.70 2.75 2.80 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

mL

2.5 3.0 3.5 4.0 4.5 5.0

En(L)/m

10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 1.0 13.0

−10−4m2Kiso

df,3 =

−10−4m2Kiso

df,3 =

ma = −10 (strongly attractive interaction) Local 3-particle interaction has significant effect

n energies, especially in region of simulations

(mL<5), and thus can be determined

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Volume-dependence of 3-body bound state

41 41

(unitary regime)

60 65 70

mL

−4 −3 −2 −1

[EB(L) − EB]/m × 105

4 5 6 7 8 9 10

mL

2.6 2.8 3.0

EB(L)/m

(a) (b) (c)

20 25 30 35 40

mL

2.96 2.97 2.98 2.99 3.00

EB(L)/m

EB(L) from q.c. EB(∞) EB(L) from q.c. EB(L) from q.c. EB(∞) ENR(L) ENR(L) ENR(L)

Need quantization condition to determine finite-volume effects for realistic values of mL

Prediction of asymptotic volume-dependence from NRQM [Meißner, Rîos, Rusetsky]

am = − 104 & m2𝒧df,3 = 2500

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Bound state wave-function

42 42

Work in unitary regime (ma=−104) and tune Kdf,3 so 3-body bound

state at EB=2.98858 m

Solve integral equations numerically to determine Mdf,3 from Kdf,3
Determine wavefunction from residue at bound-state pole
Compare to analytic prediction from NRQM in unitary limit [Hansen &

SRS, 1609.04317]

M(u,u)

df,3 (k, p) ∼ −Γ(u)(k)Γ(u)(p)∗

E2 − E2

B

. |Γ(u)(k)NR|2 = |c||A|2 256π5/2 31/4 m2κ2 k2(κ2 + 3k2/4) sin2 ⇣ s0 sinh−1 √

3k 2κ

⌘ sinh2 πs0

2

Known constant Known constant Determined by fit to volume-dependence of bound-state energy

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Bound state wave-function

43 43

|Γ(u)(k)NR|2 = |c||A|2 256π5/2 31/4 m2κ2 k2(κ2 + 3k2/4) sin2 ⇣ s0 sinh−1 √

3k 2κ

⌘ sinh2 πs0

2

0.0 0.2 0.4 0.6 0.8 1.0

k/m

10−5 10−3 10−1 101

|Γ(u)(k)|2 × 10−6

mL = 65 mL = 60 mL = 70

mL→∞ gives infinite-volume result

0-parameter prediction

Works over many orders of magnitude to expected accuracy

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Beyond isotropic: including higher partial waves

44

[Blanton, Romero-López & SRS, 1901.07095]

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Beyond the isotropic approximation

45 45

In 2-particle case, we know that s-wave scattering dominates at low

energies; can then systematically add in higher waves (suppressed by q2l)

We are implementing the same general approach for Kdf,3, making use
f the facts that it is relativistically invariant and completely symmetric

under initial- & final-state permutations, and expanding about threshold

We work in the G-parity invariant theory with 3 identical scalars, so

the first channel beyond s-wave has l=2 (d-wave)

𝒧df,3

p1 p2 p3 p′

1

p′

2

p′

3

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Beyond the isotropic approximation

46 46

leading order term:

nly term kept in

isotropic approx

nly term at linear order

Only three coefficients needed at quadratic order

m2Kdf,3 = Kiso + K(2,A)

df,3 ∆(2) A + K(2,B) df,3 ∆(2) B + O(∆3)

K K K K Kiso = Kiso

df,3 + Kiso,1 df,3 ∆ + Kiso,2 df,3 ∆2

K K K K ∆(2)

A = 3

X

i=1

(∆2

i + ∆0 2 i ) − ∆2

X ∆(2)

B = 3

X

i,j=1

e t 2

ij − ∆2

, sij ≡ (pi + pj)2 , s0

ij ≡ (p0 i + p0 j)2

, tij ≡ (pi − p0

j)2

ta. As in the two-pa

∆ ≡ s − 9m2 9m2

, ∆i ≡ sjk − 4m2 9m2

∆0

i ≡

s0

jk − 4m2

9m2

, e tij ≡ tij 9m2

𝒧df,3

p1 p2 p3 p′

1

p′

2

p′

3

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Decomposing into spectator/dimer basis

47 47

𝒧df,3

p1 p2 p3 p′

1

p′

2

p′

3

Implemented quantization condition through quadratic order, for P=0, including projection onto overall cubic group irreps

spectator momentum

}

Decompose into harmonics in dimer CM frame: l,m spectator momentum

{

l’,m’

⇒ l’=0,2 & l=0,2

For consistency, need

1 𝒧(0)

2

= 1 16πE2 [ 1 a0 + r0 q2 2 + P0r3

0q4

] 1 𝒧(2)

2

= 1 16πE2 1 q4 1 a5

2

Quadratic terms ΔA(2) & ΔB(2)

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First results including l=2

48 48

Threshold expansion works well. What happens to this level as a2 is turned on?

Results from Isotropic approximation with 𝒧df,3 = 0

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First results including l=2

49 49

δEd = [E(a2, L) − E(a2 = 0,L)]/m δEd = 294 (a0m)2(a2m)5 (mL)6 + 𝒫(a3

0 /L6,1/L7)

Determine Compare to prediction:

Works well (also for a0 and a2 dependence) Tiny effect, but checks

ur numerical

implementation

5 10 20 30 40

mL

10−13 10−12 10−11 10−10 10−9 10−8 10−7

δEd m

analytical numerical

10−11 2 × 10−11

using quantization condition

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4 5 6 7 8

mL

3.0 3.5 4.0 4.5 5.0

En(L)/m a = −1/2

First results including l=2

50 50

What happens to these levels as a2 is turned on?

Results from Isotropic approximation with 𝒧df,3 = 0

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First results including l=2

51 51

Projected onto cubic-group irrep A1+

−2.0 −1.5 −1.0 −0.5 0.0

ma2

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

E

A+ 1 n

m

E0 E1 E2 E3 E4 E5 E6

mL = 8.1, ma0 = − 0.1, r0 = P0 = 𝒧df,3 = 0

d-wave attraction can have very significant effect on energy levels

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Evidence for 3-particle bound state

52 52

ma0 = − 0.1, ma2 = − 1.3, r0 = P0 = 𝒧df,3 = 0

22 24 26 28 30 32 34 36

mL

2.874 2.875 2.876 2.877 2.878 2.879

EA+

1

m

Binding caused by d- wave attraction! Relevant for atomic physics? Quantization condition is useful as tool for studying infinite-volume!

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Impact of quadratic terms in Kdf,3

53 53

Energies of 3π+ states need to be determined very accurately to be sensitive to Kdf,3(2,B), but this is achievable in ongoing simulations

4.0 4.5 5.0 5.5 6.0 6.5 7.0

mL

0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020

∆EE+

1

m

K(2,B)

df,3 = 40

K(2,B)

df,3 = 80

K(2,B)

df,3 = 400

s- and d-wave

5 0.0134 0.0139 4.1 0.0195 0.0205

a0, r0, P0, & a2 set to physical values for 3π+

Energy shift relative to noninteracting energy for first excited state. Projected into E+ irrep.

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Numerical implementation: isotropic approximation including two-particle bound states

54

[Blanton, Briceño, Hansen, Romero-López & SRS, in progress]

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Isotropic approximation: v2

55 55

Same set-up as in [Briceño, Hansen & SRS, 1803.04169], except that by

tweaking the PV pole-prescription, the formalism works for a > 1

Allows us to study cases where, in infinite-volume, there is a two-

particle bound state (“dimer’’)

EB/m = 2 1 − 1/a2 a=2 3

Interesting case: choose parameters so that there is both a dimer and a

trimer

This is the analog (without spin) of studying the n+n+p system in which

there are neutron + deuteron and tritium bound states

The finite-volume states will have components of all three types

Preliminary

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E/m mL

Free 3-particle states Free dimer+particle states

E m = 1+ 3

Avoided level-crossing 3=trimer 2+1 1+1+1 2+1

Preliminary

Isotropic approximation: a=2, Kdf,3=0

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20 25 30 35 40 45 50 55

mL

2.68 2.70 2.72 2.74 2.76 2.78 2.80 2.82 2.84 2.86

E m

bound state ground state excited states free particle+dimer L → ∞

trimer! Dominantly 2+1 states

Preliminary

Isotropic approximation: a=2, Kdf,3=0

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−0.4 −0.2 0.0 0.2 0.4

(k/m)2

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2

k m cot δ0

bound state ground state 1st state 2nd state 3rd state 4th state fit: b0, r fit: b0, r, P E = 3m

Isotropic approximation: a=2, Kdf,3=0

58 58

2+1 EFT: solve 2-particle quant. cond. for nondegenerate particles Fit to first excited state spectrum predicts all other levels! Trimer is 2+1 bound state! E=3m

b0=6.4 2+1 phase shift 2+1 relative CM momentum

Preliminary

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−0.06 −0.04 −0.02 0.00 0.02 0.04

k2

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

k cot δ0

1st state 2nd state 3rd state 4th state 5th state 6th state 7th state ground state bound state

Isotropic approximation: a=6, Kdf,3=0

59 59

2+1 EFT: solve 2-particle quant. cond. for nondegenerate particles E=3m

b0=-4 2+1 phase shift 2+1 relative CM momentum

Very Preliminary

Trimer is probably not a 2+1 bound state! Pole reminiscent of that found in n+d and p+d spin doublet scattering

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Outlook

60

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Substantial progress on three-particle formalism
Relationship to the other methods (NREFT & FV Unitarity) now well

understood [Hansen & SRS (review)]

Freedom in PV prescription extends range of original formalism; allows

study of cases with two particle bound states and resonances [Blanton, Briceño, Hansen, Romero-López, SRS]

Similar freedom allows the use of a higher cutoff, which can be used to

investigate unphysical solutions [Blanton, Briceño, Hansen, Romero-López, SRS]

Relation of M3 to Kdf,3 provides an alternative infinite-volume description
f M3 that is unitary—may be useful in data analysis [Briceño, Hansen, SRS

& Szczepaniak]

Extensions to higher spins, nonidentical particles, multiple K-

matrix poles, and Lellouch-Lüscher factors are needed, but will likely be straightforward

Outlook

61

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The major issue is how to make the formalism practical
Need physics-based parametrizations of
Need to implement relation between and above threshold
Successful extraction of 3-body amplitude from simulations of φ4 theory

[Roméro-Lopez et al.]; application to QCD simulations is underway [HADSPEC collab.]

Moving to 4+ particles in this fashion looks challenging but

does not obviously introduce new theoretical issues

Outlook

62

𝒧df,3

ℳ3

𝒧df,3

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Backup slides

63

SLIDE 64

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Sketch of derivation

f 2-particle

quantization condition

64

[Kim, Sachrajda & SRS 05]

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Work in continuum (assume that LQCD

can control discretization errors)

Cubic box of size L with periodic BC,

and infinite (Minkowski) time

Spatial loops are sums:
Consider identical scalar particles with physical mass m, interacting arbitrarily

in a general relativistic effective field theory

Setup

65

1 L3

P

~ k

~ k = 2π

L ~

n

L

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Methodology

66

Calculate (for some P=2πnP/L)
Poles in CL occur at energies of finite-volume spectrum: consider m < E* < 3m

Full propagators Normalized to unit residue at pole Infinite-volume vertices Boxes indicated summation

ver finite-volume momenta
E.g. for 2 particles, σ ~ π2 :

CM energy is E*=√(E2-P2)

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Step 1

Replace loop sums with integrals where possible
Drop exponentially suppressed terms (~e-ML, e-(ML)^2, etc.) while keeping power-law dependence

67

Exp. suppressed if g(k) is smooth

and scale of derivatives of g is ~1/M

Possible whenever no physical, on-shell cut through loop
Can show using time-ordered PT

can replace sum with integral here

P = (E, ~ P)

but not here!

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Step 2

Use “sum=integral + [sum-integral]” if integrand has pole, using

68 q* is relative momentum

f pair on left in CM

f & g evaluated for ON-SHELL momenta Depend only on direction in CM Kinematic function

@ Z dk0 2⇥ 1 L3 X

k

− Z d4k (2⇥)4 1 A f(k) 1 k2 − m2

j + i

1 (P − k)2 − m2

j + ig(k)

= Z dΩq⇤dΩq⇤0 f ∗

j (ˆ

q∗)Fjj(q∗, q∗0)g∗

j (ˆ

q∗0)

Example

Focus on this loop k P-k

P = (E, ~ P)

g is right-hand part

f integrand

f is left-hand part

f integrand

+ exp. suppressed

g PV g PV

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Step 2

Use “sum=integral + [sum-integral]” where integrand has pole, with [KSS]

69

@ Z dk0 2⇥ 1 L3 X

k

− Z d4k (2⇥)4 1 A f(k) 1 k2 − m2

j + i

1 (P − k)2 − m2

j + ig(k)

= Z dΩq⇤dΩq⇤0 f ∗

j (ˆ

q∗)Fjj(q∗, q∗0)g∗

j (ˆ

q∗0)

Diagrammatically
ff-shell
n-shell

1 L3 X

~ k

Z

~ k

finite-volume residue

g PV g PV g PV g PV

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+ + + + · · ·

σ† σ† σ† σ†

σ σ σ σ

CL(E, ~ P) =

these loops are now integrated

Apply previous analysis to 2-particle correlator (m < E* < 3m)
Collect terms into infinite-volume Bethe-Salpeter kernels

σ†

σ

+ · · ·

σ†

σ

+

+ + · · ·

+

⇢

CL(E, ~ P) = iB

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Apply previous analysis to 2-particle correlator
Collect terms into infinite-volume Bethe-Salpeter kernels

σ†

σ

+ · · ·

σ†

σ

+

+ + · · ·

+

⇢

CL(E, ~ P) = + + · · · +

σ†

σ

σ†

σ

σ†

σ

CL(E, ~ P) =

Leading to

iB iB iB iB

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A0

⇢ ⇢ + + · · · σ σ ⇢ + · · ·

F

iB iB

+ ...

72

+ + · · · +

σ†

σ

σ†

σ

σ†

σ

+

σ† σ σ† σ

CL(E, ~ P) =

σ† σ σ† σ σ† σ σ† σ

+ + +

F F F F F

Next use sum identity

A CL(E, ~ P) = C∞(E, ~ P) +

⇢ + σ†

σ†

zero F cuts matrix elements:

And regroup according to number of “F cuts”

iB iB iB

iB iB iB iB

ne F cut

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⇢ ⇢ + + · · ·

+ · · · +

iM A0

A CL(E, ~ P) = C∞(E, ~ P) +

two F cuts

A0

A

F F F the infinite-volume, on-shell 2→2 scattering amplitude

And keep regrouping according to number of “F cuts”

73

+ + · · · +

σ†

σ

σ†

σ

σ†

σ

+

σ† σ σ† σ

CL(E, ~ P) =

σ† σ σ† σ σ† σ σ† σ

+ + +

F F F F F

Next use sum identity

iB iB iB

iB iB iB iB

iB iB iB

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⇢ ⇢ + + · · ·

+ · · · +

A0

A CL(E, ~ P) = C∞(E, ~ P) +

A0

A

F F F the infinite-volume, on-shell 2→2 K-matrix

Alternate form if use PV-tilde prescription:

74

+ + · · · +

σ†

σ

σ†

σ

σ†

σ

+

σ† σ σ† σ

CL(E, ~ P) =

σ† σ σ† σ σ† σ σ† σ

+ + +

F F F F F

Next use sum identity

iB iB iB

iB iB iB iB

iB iB iB

g PV

g PV g PV

iK

g PV g PV g PV g PV g PV

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Final result:

+ + + + · · ·

iM iM iM A0 A0 A0

A A A CL(E, ~ P) = C∞(E, ~ P) CL(E, ~ P) = C1(E, ~ P) +

1

X

n=0

A0iF[iM2!2iF]nA

F F F F F F

Correlator is expressed in terms of infinite-volume, physical quantities and

kinematic functions encoding the finite-volume effects

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diverges whenever diverges
76
Final result:

+ + + + · · ·

iM iM iM A0 A0 A0

A A A CL(E, ~ P) = C∞(E, ~ P) CL(E, ~ P) = C1(E, ~ P) +

1

X

n=0

A0iF[iM2!2iF]nA

F F F F F F

CL(E, ~ P) = C1(E, ~ P) + A0iF 1 1 − iM2!2iF A

no poles,

nly cuts
no poles,
nly cuts

matrices in l,m space

iF 1 1 − iM2→2iF CL(E, ~ P)

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77
Final result:

+ + + + · · ·

iM iM iM A0 A0 A0

A A A CL(E, ~ P) = C∞(E, ~ P) CL(E, ~ P) = C1(E, ~ P) +

1

X

n=0

A0iF[iM2!2iF]nA

F F F F F F

CL(E, ~ P) = C1(E, ~ P) + A0iF 1 1 − iM2!2iF A

no poles,

nly cuts
no poles,
nly cuts

matrices in l,m space

∆L, ~

P (E) = det

⇥ (iF)−1 − iM2→2 ⇤ = 0 ⇒ ∆L, ~

P (E) = det

⇥ (Fg

P V )−1 + K2

⇤ = 0 ⇒

Alternative form

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Sketch of derivation of 3-particle quantization condition

78

[Hansen & SRS, arXiv:1408.5933 & 1504.04248]

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Derivation

79 79

Generic relativistic EFT, working to all orders
Do not need a power-counting scheme
To simplify analysis: impose a global Z2 symmetry (G parity) & consider identical scalars
Obtain spectrum from poles in finite-volume correlator
Consider ECM < 5m so on-shell states involve only 3 particles

Momentum sums rather than integrals Infinite-volume Bethe-Salpeter kernels Arbitrary

perator

creating 3 particles

(1)

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Derivation

80 80 Momentum sums rather than integrals Infinite-volume Bethe-Salpeter kernels Arbitrary

perator

creating 3 particles

Replace sums with integrals plus sum-integral differences to extent possible
If summand has pole or cusp then difference ~1/Ln and must keep (Lüscher zeta function)
If summand is smooth then difference ~ exp(-mL) and drop
Avoid cusps by using PV prescription—leads to generalized 3-particle K matrix
Subtract above-threshold divergences of 3-particle K matrix—leads to Kdf,3

(2)

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Derivation

81 81

Reorganize, resum, … to separate infinite-volume on-shell relativistically-invariant

non-singular scattering quantities (K2, Kdf,3) from known finite-volume functions (F [Lüscher zeta function] & G [“switch function”])

(3)

det [F−1

3

+ 𝒧df,3]

⇒

= 0

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Derivation

82 82

Relate Kdf,3 to M3 by taking infinite-volume limit of finite-volume scattering amplitude
Leads to infinite-volume integral equations involving M2 & cut-off function H
Can formally invert equations to show that Kdf,3 (while unphysical) is relativistically

invariant and has same properties under discrete symmetries (P , T) as M3

(4)

Involve only M2 and G so “known” Sums over k go over to integrals with iε pole prescription